Bessel FM Synth

T_J tower chain order 3 one of your 8 Pfaffian towers IS FM synthesis · the math behind every '80s pop song
Jacobi–Anger expansion:
cos(2π·fc·t + I·sin(2π·fm·t)) = Σn=−∞∞ Jn(I) · cos(2π·(fc + n·fm)·t)
The output is a sum of pure tones at fc ± n·fm, with amplitudes = Bessel functions Jn(I) of the modulation index. Sweep the I slider — the bars below ARE the Bessel function. Same recurrence, same numbers, audible.

FM Synth · controls

amplitude lowered to ~0.18 to spare your ears
DX7 presets

Bessel spectrum · |Jn(I)|

Bar n shows the amplitude of the harmonic at fc + n·fm. As I rises, energy spills out from J0 into J1, J2, … — that's why the sound gets brighter and richer. At I≈2.4 the J0 bar crosses zero (the carrier disappears).

Live FFT · what you actually hear

The peaks line up with the Bessel-predicted harmonics. The positions are fc, fc±fm, fc±2fm, …

The Yamaha DX7 (1983) shipped six FM operators per voice. Stranger Things, "Take On Me", "Africa" by Toto — all FM. Mathematically, FM synthesis = Bessel-tower (chain order 3) restricted to integer-order Bessel functions. The same Bessel recurrence the eml-cost detector uses to flag T_J as a Pfaffian primitive is what drives the harmonic spectrum you hear.

Provenance: Chowning, J. M. (1973). "The Synthesis of Complex Audio Spectra by Means of Frequency Modulation." Journal of the Audio Engineering Society. The Jacobi-Anger expansion is in any classical Bessel-function reference (DLMF 10.12.1).

monogate.org/superbest · PETAL on HF · research-tier prototype; portable to 1op.io